552 XI. HYDRODYNAMICS. 



where a, Z>, c, d are constants. If we determine them so that the 

 velocity vanishes for planes at a distance +h from the X-axis, we have 



238) u = ^(f-V). 



The amount of liquid that flows through such a laminar tube per 

 unit of width parallel to the Z-axis is accordingly 



239) 



h 

 and for a length of tube I the difference of pressures at the ends is 



240) A - ft -aJ, 3=<l!Lp>, 



so that the flow is proportional to the difference of pressures at the 

 ends and inversely to the viscosity. 



For the practical determination of viscosity, we may take the 

 almost equally simple case of cylindrical flow, where the velocity has 

 everywhere the same direction, and depends upon the distance r from 

 the axis of a circular tube, at the surface of which it is at rest. 



If we put u = v = we have the equations of motion and of 

 continuity 



o^i\ dp 



241) 



and since w depends only on r the first becomes 



C*AO\ , dp 



242) 



where a is a constant as before. This equation is integrated as in 

 182, 58'), 



243) w = ^-r 2 + fclogr -f c. 



Since w is finite when r = we must have b = and if w vanishes 

 for r = R we obtain 



244) (,. 

 For the flow we find 



ctAc.\ s\ 



245) Q = 







This method was invented by Poiseuille 1 ) for the measurement of 



1) Poiseuille, Eecherches experimentales sur le mouvement des liquides dans 

 les tubes de tres petits diametres. Comptes Rendus, 1840 41; Mem. des Savants 

 Strangers, t. 9, 1846. 



